Unit 7 Trigonometric Identities and Equations 7.1 Exploring Equivalent Trig Functions When we look at the graphs of sine, cosine, tangent and their reciprocals, it is clear that there will be points where two or more of them intersect. This represents a point where trig functions are equal. We will be exploring this today. Odd & Even Functions i) The Sine Function F(x) = sin(x) is an odd function Therefore, f(-x) = -sin(x) ii) The Cosine Function F(x) = cos(x) is an even function Therefore, f(-x) = cos(x) iii) The Tangent Function F(x) = tan(x) is a(n) function How do we tell? We look at the angle (-x) which is actually 2pi x How are Sine and Cosine Related? Graph the sine function, what transformation will turn it into the cosine function? Graph the cosine function, what transformation will turn it into the sine function? (Answers are: cos(x pi/2) = sin(x) = sin(x + pi/2) = cos(x)
Relating a Function to Its Period Sine Function, period = 2pi Sin(x + 2pi) = Cosine Function, period = 2pi Cos(x + 2pi) = Tangent Function, period = Tan(x + ) = Relationships between Complimentary Angles (Cofunction Identites) Complimentary angles are (x) and (pi/2 x) The easiest way to explore the relationship is by setting up a right angle triangle (find sine and cosine for both x, and pi/2 x and compare) also find for tan(x) and tan(pi/2 x) using sine and cosine A x C pi/2 - x B Tan (x) = sin(x) tan(pi/2 x) = sin(pi/2 x) = cos(x) = cot(x) Cos (x) cos(pi/2 x) sin(x) Relationships between Angles in Different Quadrants Assume (x) is the principal angle in quadrant I, how does this relate to the related acute angles in each quadrant? Show using cart. plane Quadrant 2 = pi x Quadrant 3 = pi + x Quadrant 4 = 2pi x
Example: Pg. 393 #5a Write an expression that is equivalent to sin(7pi/8) using the related acute angle Sin(7pi/8) is in the second quadrant, and is therefore positive The only other quadrant sine is positive in is the first quadrant and the related acute angle would be sin(pi/8) Example: Pg. 393 #7a State whether cos(x + 2pi) = cos (x). Justify your decision. The period for cosine is 2pi, therefore x + 2pi is equal to x +
7.2 Compound Angle Formulas An angle that is created by adding or subtracting two or more angles When are compound angles used? To obtain exact values for trigonometric rations (break angles into angles from special triangles) To show that trig ratios are equivalent To prove trig identities (7.4) To solve trig equations Compound Angle Formulas Addition Formulas sin(a + b) = sin a cos b + cos a sin b cos(a + b) = cos a cos b sin a sin b tan a + tan b tan(a + b) = 1 tan a tan b Subtraction Formulas sin(a b) = sin a cos b cos a sin b cos(a b) = cos a cos b + sin a sin b tan a tan b tan(a + b) = 1 + tan a tan b Example: Rewrite sin(2a)cos(a) + cos(2a)sin(a) as a single trigonometric ratio. Sin(2a + a) or sine(3a) Example: Determine the exact value of cos(15 ) Cos(15) = cos(45 30) = cos(45)cos(30) + sin(45)sin(30) = ( 2/2)( 3/2) + ( 2/2)(1/2) = 6 + 2 4 Example: Simplify cos(7pi/12)cos(5pi/12) - sin(7pi/12)sin(5pi/12) = cos (7pi/12 + 5pi/12) = cos (pi) = -1
Example: Evaluate sin(x + y) if sin x = 4/5 and sin y = -12/13, where 0 < x < pi/2, 3pi/2 < y < 2pi Sin(x + y) = sin(x)cos(y) + cos(x)sin(y) Opp = 4 Opp = -12 Hyp = 5 Hyp = 13 Adj = (5 2 4 2 ) Adj = (13 2 12 2 ) = (25 16) = (169 144) = 9 = 25 = 3 = 5 Cos(x) = 3/5 Cos(y) = 5/13 Sin(x + y) = (4/5)(5/13) + (3/5)(-12/13) = 20/65 36/65 = -16/65 Try Pg. 400 #1-4,5abc,6,8cde,9-13,15
7.3 Double Angle Formulas Show how a double angle (2x) relate to the original angle (x) Can all be derived from the compound angle formulas Example: Use the compound angle formula for sine to show that Sin(2x) = 2sin(x)cos(x) Sin(2x) = sin(x + x) = sin(x)cos(x) + cos(x)sin(x) = 2sin(x)cos(x) Example: Use the compound angle formula to derive the double angle formula for cosine and tangent: Cos(2x) = cos(x + x) = cos(x)cos(x) sin(x)sin(x) = cos 2 x sin 2 x Given that sin 2 x + cos 2 x = 1 = cos 2 x (1 cos 2 x) OR = (1 sin 2 x) sin 2 x Cos(2x) = 2cos 2 x 1 OR = 1 2sin 2 x Tan(2x) = tan(x) + tan(x) 1 tan(x)tan(x) = 2tan(x) 1 tan 2 x Example: Write 2sin4xcos4x as a single trigonometric ratio. Sin(2x) = 2sin(x)cos(x) 2sin(4x)cos(4x) in this case x = 4x Sin(2(4x)) = 2sin(4x)cos(4x) Sin(8x) = 2sin(4x)cos(4x)
Example: Express 1 2sin 2 (3pi/8) as a single trig ratio and evaluate Cos(2x) = 1 2sin 2 x in this case x = 3pi/8 Cos(2(3pi/8)) = Cos(3pi/4) with related acute angle in Q2 (pi/4) Cos(3pi/4) = 2/2 Example: Use the double angle formula to rewrite tan6x Tan(6x) = Tan2(3x) = 2tan(3x) 1 tan 2 (3x) Example: Determine the values of sin2x and cos2x, given cos x = -2/3 and 0 x pi Adj = -2 (therefore in Q2) Hyp = 3 Opp = (3 2 2 2 ) = 5 (positive in Q2) Sin(x) = 5/3 Sin(2x) = 2sin(x)cos(x) = 2( 5/3)(-2/3) = -4 5 9 Cos(2x) = 1-2sin 2 x = 1 2( 5/3) 2 = 1 2(5/9) = 1 10/9 = -1/9 Try Pg. 407 #1-11,15a,16a,17